Explaining the size of tree leaves using physics

Tall trees get limited by a maximum leaf size.

One of the most useless-but-cool things about physics is the post-hoc information it offers for the field of biology. I suspect that these physics-focused explanations really piss biologists off, and rightly so, since they tend to offer an explanation for observations, but have no predictive power to point us beyond what we already know.

Biology is a big field with lots of sub-fields, though. For the most part, biologists interested in mammals and birds have had to bear the burden of physicists' attention. But that's changing, as plant biologists have become the latest victims. Yes, physics now offers an explanation for why tall trees have such, ahem, tiny leaves.

Before we get to the trees and their pitifully small leaves, let's take a quick look at the sort of arguments and insights that physics offers biologists. For instance, it can help you determine why the smallest and largest land-dwelling mammals are about the size that they are—look no further than the balance of energy. Mammals are warm-blooded, and that costs some energy. And, to make matters worse, they lose energy through their skin simply through heat conduction. The smaller you are, the greater the proportion of your energy is spent maintaining your internal body temperature, simply because you have more surface area relative to body mass. At a certain size, you simply cannot eat fast enough to maintain an internal body temperature.

On the upper end of the scale, you have many more cells to supply with energy, so you spend more energy foraging for food. At some point, you end up spending more energy on foraging than the total available energy in the food being foraged. Hence, at the bottom of the scale shivers the shrew, scurrying around desperately eating to keep warm, while at the upper end, an elephant is constantly and slowly eating everything in sight.

It is possible to make similar arguments based on the strength of bone vs. total mass, blood pressure for height, and metabolic rate for average lifetime. Now, I know I made this look a bit of a joke in the introduction, because, truth be told, these sorts of arguments are fun. In all seriousness, though, this form of reasoning—figuring out limitations due to the way that physical processes scale with increases and decreases in particular parameters—is very handy. More fascinating is that they are at least broadly in agreement with observations.

Resistance makes growth futile

Now a pair of scientists have applied a scaling law arguments to the leaves of trees. The heart of the argument is about how fast sugar flows down the trunk of the tree. A tree has to be able to distribute energy from the leaves to the roots, and this is all governed by flow through a network of tubes, called phloem.

The speed of the flow is in turn governed by just a few factors: two different flow resistances, and the pressure difference between the top and the bottom of the tree. It turns out that the pressure difference is independent of height, but the flow resistances are a different matter.

Flow resistance in the trunk depends on the length and radius of the phloem tube. As the length increases, the flow resistance increases. But, as the radius increases, the resistance drops. For small trees, both the radius and the length increase with height, so the flow resistance may drop as the tree grows. Unfortunately, for taller trees, the radius tops out at 20 micrometers, leaving the flow resistance to continue increasing as the tree grows.

For leaves, the story is a little more complicated. The phloem has permeable membranes that are designed to accept increasing amounts of sugar from the leaf. The argument is that the increased permeability and phloem surface area result in greater concentration differences that drive a higher pressure differential within the leaf. Looking at the leaf as a black box, it simply appears that the internal resistance has dropped. The upshot is that the flow resistance of the leaf decreases as the leaf size increases.

Why does this matter? Both resistances contribute to slowing the flow but, if one is much larger than the other, then changing the smaller resistance has no impact. So, for a tall tree, increasing the leaf size doesn't result in faster flow, since the phloem's resistance dominates. For smaller trees, a larger leaf size does result in faster flow. And this changes the energy balance for the tree.

My, what expensive leaves you have

The speed of energy flow has a maximum when the leaf resistance is zero, but that maximum is approached very, very slowly. At some point, the cost of maintaining the larger leaf is greater than the marginal increase in energy flow. Once you choose a tree height and a cost factor, you end up with a maximum leaf size.

On the lower end of the scale, the tree requires a minimum flow of energy. This is governed by cell-to-cell diffusion of sugar-water. Once you plug this number in, for any given tree height, a minimum leaf size pops out.

As the tree grows taller, the minimum leaf size goes up, while the maximum leaf size goes down. At around 100m, the two cross, so that the minimum required leaf size is greater than the maximum useful leaf size. And, guess what, the tallest known trees top out at around 100m. More importantly though, the variation in leaf size decreases with increasing tree height, and this is exactly what the observational data shows.

In fact, the data fits remarkably well. For the largest leaf size, there is a cost-factor that they have to fit. But, considering that a single parameter fits the data for multiple species within the angiosperm family (flowering plants; the study didn't consider anything else), that is pretty good. The lower limit on leaf size is a lot rougher, but the data is also a lot noisier at the lower bound, so no simple curve fit that data well.

What I love about papers like this is how a few simple physical principles come together to provide a rough explanation for observed data. What makes it more interesting is that it sets criteria that can help us find exceptions, and exceptions make for interesting research.

Promoted Comments

One of the most useless-but-cool things about physics are the post-hoc explanations it offers for the field of biology. I suspect that these physics-focused explanations really piss biologists off, and rightly so, since they tend to offer an explanation for observations, but have no predictive power to point us beyond what we already know.

But... you said at the end of the article that these flow resistances mean trees shouldn't grow taller than ~100m, which sure sounds like a prediction (it just happens to be already well-observed by botanists). And you said this kind of work can aid in finding exceptions to the rules, which seems to me as "beyond what we already know".

So I'm not sure what this paragraph is telling us if you contradict it later in the story.

This model cannot predict the exceptions to the rule since, by definition, the model IS the rule. Therefore, it's descriptive if it appears to apply across a broad range of species but tree heights and leaf size vs. tree heights have already been categorized. That such a simple model seems to capture the upper end of both distributions is kinda neat. If a tree species is identified whose leaves are substantially smaller than the minimum size or larger than the maximum size given the height then there's something cool to look at. But, once again, the current model will be no help in explaining the exception except as a starting point of what's NOT going on. Nor can you predict that exceptions will exist.

What makes it more interesting is that it sets a criteria that can help us find exceptions, and exceptions make for interesting research.

This. If the simple physical argument holds for numerous cases then the outlier must do something differently than the rest. and since you know what physical principles you based your model on you also know where to start looking.

Nice, and as a biologist, I kind-of rely on that fundamentally underlying physics-y thing so I don't feel too grumpy about it. I wonder whether the model would be even better if they took into consideration water flow from roots upwards to the leaves. I recall from college that this was affected by transpiration rate (loss of water from the leaves), capillarity (in turn partly dependent on the thickness of the xylem tubes that span the height of the tree) and root pressure. Presumably leaf size is dependent on the amount of water it can get from the roots, and also determines, partly, the water flow rate?

What makes it more interesting is that it sets a criteria that can help us find exceptions, and exceptions make for interesting research.

This. If the simple physical argument holds for numerous cases then the outlier must do something differently than the rest. and since you know what physical principals you based your model on you also know where to start looking.

One of the most useless-but-cool things about physics are the post-hoc explanations it offers for the field of biology. I suspect that these physics-focused explanations really piss biologists off, and rightly so, since they tend to offer an explanation for observations, but have no predictive power to point us beyond what we already know.

But... you said at the end of the article that these flow resistances mean trees shouldn't grow taller than ~100m, which sure sounds like a prediction (it just happens to be already well-observed by botanists). And you said this kind of work can aid in finding exceptions to the rules, which seems to me as "beyond what we already know".

So I'm not sure what this paragraph is telling us if you contradict it later in the story.

One of the most useless-but-cool things about physics are the post-hoc explanations it offers for the field of biology. I suspect that these physics-focused explanations really piss biologists off, and rightly so, since they tend to offer an explanation for observations, but have no predictive power to point us beyond what we already know.

But... you said at the end of the article that these flow resistances mean trees shouldn't grow taller than ~100m, which sure sounds like a prediction (it just happens to be already well-observed by botanists). And you said this kind of work can aid in finding exceptions to the rules, which seems to me as "beyond what we already know".

So I'm not sure what this paragraph is telling us if you contradict it later in the story.

This model cannot predict the exceptions to the rule since, by definition, the model IS the rule. Therefore, it's descriptive if it appears to apply across a broad range of species but tree heights and leaf size vs. tree heights have already been categorized. That such a simple model seems to capture the upper end of both distributions is kinda neat. If a tree species is identified whose leaves are substantially smaller than the minimum size or larger than the maximum size given the height then there's something cool to look at. But, once again, the current model will be no help in explaining the exception except as a starting point of what's NOT going on. Nor can you predict that exceptions will exist.

As a physicist, I appreciate the analysis and how it fits the data. However, I have to ask the question: where is the feedback loop that controls this process? Trees don't have intelligence (at least that we know of), so there has to be a biochemical feedback loop that stops growing the leaves once they're in the optimal size range. Moreover, the tree doesn't even "know" what the optimal size range is, so the feedback loop has to be controlled by an external variable (sunlight, energy from the sugars, etc). Consider the mechanism behind plant phototropism as an example of such a loop.

So while the physics of determine the optimal leaf size is interesting, I'm now much more interested in understanding how trees actually zero in on the optimal size for each tree.

As a physicist, I appreciate the analysis and how it fits the data. However, I have to ask the question: where is the feedback loop that controls this process? Trees don't have intelligence (at least that we know of), so there has to be a biochemical feedback loop that stops growing the leaves once they're in the optimal size range. Moreover, the tree doesn't even "know" what the optimal size range is, so the feedback loop has to be controlled by an external variable (sunlight, energy from the sugars, etc). Consider the mechanism behind plant phototropism as an example of such a loop.

So while the physics of determine the optimal leaf size is interesting, I'm now much more interested in understanding how trees actually zero in on the optimal size for each tree.

It's evolution. Trees with bigger and smaller leaves were less fit, and the ones that are left have genes coding for the proper size. I'm no biologist, but I'd guess that the genes that control the size are close analogues of the ones that control things like skeletal shape in animals.

As a physicist, I appreciate the analysis and how it fits the data. However, I have to ask the question: where is the feedback loop that controls this process? Trees don't have intelligence (at least that we know of), so there has to be a biochemical feedback loop that stops growing the leaves once they're in the optimal size range. Moreover, the tree doesn't even "know" what the optimal size range is, so the feedback loop has to be controlled by an external variable (sunlight, energy from the sugars, etc). Consider the mechanism behind plant phototropism as an example of such a loop.

So while the physics of determine the optimal leaf size is interesting, I'm now much more interested in understanding how trees actually zero in on the optimal size for each tree.

It's evolution. Trees with bigger and smaller leaves were less fit, and the ones that are left have genes coding for the proper size. I'm no biologist, but I'd guess that the genes that control the size are close analogues of the ones that control things like skeletal shape in animals.

So would this mean that we could roughly predict what size a tree would become, simply by looking at the leaf size of the young plant? (I guess such a tool would be useless, but still interesting.)

I recall from college that this was affected by transpiration rate (loss of water from the leaves), capillarity (in turn partly dependent on the thickness of the xylem tubes that span the height of the tree) and root pressure.

I though it was this phenomenon (capillarity + evaporation) that limited the size of trees: the leaves pull the water up, gravity pulls the water down, if your liquid column gets too big, the water make bubbles spontaneously (and doesn't goes any upper).

So if multiple phenomenon limit the size of trees, maybe not all of them are causes and some are consequences. For example, why develop complicated leaves that work above 100m if your xylem tubes won't work above 80m? (it work both ways)

I recall from college that this was affected by transpiration rate (loss of water from the leaves), capillarity (in turn partly dependent on the thickness of the xylem tubes that span the height of the tree) and root pressure.

I though it was this phenomenon (capillarity + evaporation) that limited the size of trees: the leaves pull the water up, gravity pulls the water down, if your liquid column gets too big, the water make bubbles spontaneously (and doesn't goes any upper).

So if multiple phenomenon limit the size of trees, maybe not all of them are causes and some are consequences. For example, why develop complicated leaves that work above 100m if your xylem tubes won't work above 80m? (it work both ways)

Or xylem tubes max out at 20 microns because of the reasons you state and the max tree height is a consequence of that. Another reason this is a descriptive model rather than predictive.

Completely fallacious, but I wonder how this could be applied to explain the Kashyyyk trees Wookiees live in. Those things hold entire cities.

This research indicates that such trees absolutely must have pumps and not rely on capillary action in their phloem, or else have an entirely different mechanism for creating and transporting chemical energy.

One of the most useless-but-cool things about physics are the post-hoc explanations it offers for the field of biology. I suspect that these physics-focused explanations really piss biologists off, and rightly so, since they tend to offer an explanation for observations, but have no predictive power to point us beyond what we already know.

But... you said at the end of the article that these flow resistances mean trees shouldn't grow taller than ~100m, which sure sounds like a prediction (it just happens to be already well-observed by botanists). And you said this kind of work can aid in finding exceptions to the rules, which seems to me as "beyond what we already know".

So I'm not sure what this paragraph is telling us if you contradict it later in the story.

This model cannot predict the exceptions to the rule since, by definition, the model IS the rule. Therefore, it's descriptive if it appears to apply across a broad range of species but tree heights and leaf size vs. tree heights have already been categorized. That such a simple model seems to capture the upper end of both distributions is kinda neat. If a tree species is identified whose leaves are substantially smaller than the minimum size or larger than the maximum size given the height then there's something cool to look at. But, once again, the current model will be no help in explaining the exception except as a starting point of what's NOT going on. Nor can you predict that exceptions will exist.

Actually, you can predict exceptions from a model. For if the model is correct, then any exceptions will be the result of simplifications of the model--assumptions if you will--.

Here are a few possibilities: Larger phloem, faster sugar transfer due to specialized proteins/oils/fats, different sugar type, not using sugar, but oils and fats, "localized" sugar generation (parts of the tree being relatively stand-alone), anything not included in the model or assumed to be nearly the same.

the same principle applies to the bone argument: the model only works if you follow the assumptions: that the bone is calcium. What if the bone is based off of nanotube structures? or based off of weak metals? or just cartilage instead? These all 'break' the model, but not really. You can find them by looking at assumptions within the model.

This is why mammals use fats and oils and a non-viscous bloodstream: because it is much faster and can move more energy faster. You dont see plants running around, do you?

This model cannot predict the exceptions to the rule since, by definition, the model IS the rule. Therefore, it's descriptive if it appears to apply across a broad range of species but tree heights and leaf size vs. tree heights have already been categorized. That such a simple model seems to capture the upper end of both distributions is kinda neat. If a tree species is identified whose leaves are substantially smaller than the minimum size or larger than the maximum size given the height then there's something cool to look at. But, once again, the current model will be no help in explaining the exception except as a starting point of what's NOT going on. Nor can you predict that exceptions will exist.

The model as defined may not be able to predict exceptions that are true outliers but I wonder if simple statistical methods can tell us more. How about this as a suggestion;- Given that the current model defines the upper and lower limits with some known assumptions, can a data-gathering exercise look into tree heights and leaf sizes and compile enough data to form a "good" statistical distribution?- Then, by fitting the above distribution, can we compare the distribution of real cases to the expected bounds and estimate the probability of occurrence of specimens outside these bounds?- Knowing the distribution may also tell us if it is Gaussian (as one would expect things in nature to be) or something else (maybe even multimodal?)

Phloem, phlegm, ... How come english has all the fün words? I feel funnily disadvantaged. :-/

Wickwick wrote:

This model cannot predict the exceptions to the rule since, by definition, the model IS the rule. Therefore, it's descriptive if it appears to apply across a broad range of species but tree heights and leaf size vs. tree heights have already been categorized. That such a simple model seems to capture the upper end of both distributions is kinda neat. If a tree species is identified whose leaves are substantially smaller than the minimum size or larger than the maximum size given the height then there's something cool to look at. But, once again, the current model will be no help in explaining the exception except as a starting point of what's NOT going on. Nor can you predict that exceptions will exist.

This description, which appears often, is like descriptions of "verification" et cetera, perhaps somewhat myopically centered in the research "now".

In the process view of science, what a fully tested theory lacks is further tests. If it then has potential contenders, there is too little constraint.

A similar situation appeared for, say, absolute vs relative space in classical mechanics. It wasn't until special relativity was developed that the degeneracy was broken. Even if for simplicity, classical mechanics abandoned absolute space between Newton's mechanics and Lagrange's.

But I don't think either classical mechanics were seen as post hoc, the first was seen as tested and the latter as a valid reformation.

I think that the study only refers to angiosperms, not gymnosperms, and that's an important distinction.

Predictions? What's the leaf size for top leaves in young trees exposed to the sun? I think that the model would say that they would be larger than top leaves in the old trees. BTW that's true for needle *length* for Sequoia sempervirens but I have no idea about area or volume.

Completely fallacious, but I wonder how this could be applied to explain the Kashyyyk trees Wookiees live in. Those things hold entire cities.

This research indicates that such trees absolutely must have pumps and not rely on capillary action in their phloem, or else have an entirely different mechanism for creating and transporting chemical energy.

Please don't send that thought to Disney. I'm not sure I want a dissection of Kashyyyk trees and how midichlorians run the "pumps" in the next 40 films they'll pump out.

Biology is eminantly interesting. It never occurred to me to ask "why don't trees keep growing?" I feel like that's a child's question, which is exactly the sort of thing that I should be asking as someone who wants to know how the world works. It's a very important part of science. Keep asking questions just like a three-year-old does.

First, we assume the tree is a sphere, since that makes the math simpler.

Thanks to Moore's Law we can now consider trees to be cylinders instead!

I laughed at the first comment a lot.

This one set me off again.

I do wonder at how true this is though - the tallest trees in the world are redwoods (which have "needles", which are very small leaves) while the third tallest does indeed have proper leaves, but I suspect of a very different size from redwood needles.

I can give you several examples of VERY tall trees where this just isn't so. One being the Breadfruit tree. Another is the Durian. Seed grown avocado trees go straight up and have broad, large leaves. Sapote, particularly the Chocolate Sapote has leaves up to 12 inches long. And I can show you photos of these right off my families farm on the Big Island of Hawaii.

My thinking is, if there are several exceptions to the "rule", then the rule isn't much good for anything short of lumping some trees into genera and saying that the rule only holds true for "some" trees, in a certain grouping. Even so, its not a good rule if it only holds true for the small leaved, tall trees that were studied. If you don't study ALL tall trees, then…. it ain't science.

I found this article quite interesting although not completely easy to follow.

My question is does the writer think scientists believe that sugar transport is entirely passive, affected only by osmotic gradient differences, gravity and capillary forces or are their possibly active transport forces occurring? For instance, in my speculations regarding plant hormones, http://planthormones.info/, I explain that Auxin and probably Jasmonate are actively transported downward. Auxin is known to attract sugar to wherever it is found, so if you actively transport auxin down, sugar should follow. The same I think is true for Jasmonate.

So how would this affect the arguments presented if it were true?

Also by the way following some of the speculations in the comments, I believe xylem flow upwards is not totally explained by capillary action and evaporation force pulling up water from below when water evaporates from leaves i.e. water molecules being pulled up to replace the ones evaporating into the air. I do believe there are little pump in most plants. In this scheme the pump handles would be the leaves, the pumping action is provided by the wind fluttering the leaves up and down. All that is needed in this mechanism is a one way valve like system in the leaf stems or in the junction between the leaf and the branch.

The phenomenon known as Epinasty which often occurs in flooded plants, where the plant deliberately grows its leaves so they droop down, would be the attempt to pick up more wind action to pump more water out from the roots. Epinasty is produced by Ethylene, but also probably by Salicylic Acid which is found in great abundance in Weeping Willow tree bark, which is probably an attempt by the Willow to keep its roots dry, pumping out water with the up and down motion of the weeping willow leaves in wind as Willows are often found growing on soggy river banks.

Completely fallacious, but I wonder how this could be applied to explain the Kashyyyk trees Wookiees live in. Those things hold entire cities.

This research indicates that such trees absolutely must have pumps and not rely on capillary action in their phloem, or else have an entirely different mechanism for creating and transporting chemical energy.

No, the great height of Wroshyr trees on Kashyyyk was due to runaway terraforming by the Rakatan Empire long before the Galactic Republic was formed

Chris Lee / Chris writes for Ars Technica's science section. A physicist by day and science writer by night, he specializes in quantum physics and optics. He lives and works in Eindhoven, the Netherlands.